反问题估计的Needlet算法

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摘要翻译：
本文提出了一种新的反问题处理算法，该算法将传统的奇异值分解反问题与适当的阈值化技术相结合，选择了一个合适的新基。我们的目标是设计一种既具有小波分解的局部化和多尺度分析的优点，又不损失SVD分解的稳定性和可计算性的反演方法。为此，我们利用在奇异值分解基础上构建的局部框架（称为“针”）。我们考虑两种不同的情形：“小波”情形，假定针的行为类似于真小波；“雅可比型”情形，假定框架的性质确实依赖于手边的SVD基（因此依赖于算子）。为了说明每种情况，我们分别将估计算法应用于反卷积问题和Wicksell问题。在后一种情况下，当SVD基为Jacobi多项式基时，我们证明了我们的格式能够在$L_2$情况下获得最优的收敛速度，我们获得了文献中新的（就我们所知的）其他$L_P$范数的有趣的收敛速度，并给出了仿真研究，表明NEED-D估计器在几乎所有情况下都优于其他标准算法。
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英文标题：
《Needlet algorithms for estimation in inverse problems》
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作者：
G\'erard Kerkyacharian, Pencho Petrushev, Dominique Picard, Thomas
  Willer
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最新提交年份：
2007
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分类信息：

一级分类：Mathematics        数学
二级分类：Statistics Theory        统计理论
分类描述：Applied, computational and theoretical statistics: e.g. statistical inference, regression, time series, multivariate analysis, data analysis, Markov chain Monte Carlo, design of experiments, case studies
应用统计、计算统计和理论统计：例如统计推断、回归、时间序列、多元分析、数据分析、马尔可夫链蒙特卡罗、实验设计、案例研究
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一级分类：Statistics        统计学
二级分类：Statistics Theory        统计理论
分类描述：stat.TH is an alias for math.ST. Asymptotics, Bayesian Inference, Decision Theory, Estimation, Foundations, Inference, Testing.
Stat.Th是Math.St的别名。渐近，贝叶斯推论，决策理论，估计，基础，推论，检验。
--

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英文摘要：
  We provide a new algorithm for the treatment of inverse problems which combines the traditional SVD inversion with an appropriate thresholding technique in a well chosen new basis. Our goal is to devise an inversion procedure which has the advantages of localization and multiscale analysis of wavelet representations without losing the stability and computability of the SVD decompositions. To this end we utilize the construction of localized frames (termed "needlets") built upon the SVD bases. We consider two different situations: the "wavelet" scenario, where the needlets are assumed to behave similarly to true wavelets, and the "Jacobi-type" scenario, where we assume that the properties of the frame truly depend on the SVD basis at hand (hence on the operator). To illustrate each situation, we apply the estimation algorithm respectively to the deconvolution problem and to the Wicksell problem. In the latter case, where the SVD basis is a Jacobi polynomial basis, we show that our scheme is capable of achieving rates of convergence which are optimal in the $L_2$ case, we obtain interesting rates of convergence for other $L_p$ norms which are new (to the best of our knowledge) in the literature, and we also give a simulation study showing that the NEED-D estimator outperforms other standard algorithms in almost all situations.
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PDF链接：
https://arxiv.org/pdf/705.0274

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关键词：Need let localization Multivariate Presentation 框架 Jacobi problem 构建 收敛